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Smaller generating pair sets for RMS congruences #457

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Mar 21, 2018
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Smaller generating pair sets for RMS congruences #457

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3c0e687
tests: make conguniv.tst less specific
mtorpey Feb 14, 2018
d7587bd
rmscong: reduce number of generating pairs returned
mtorpey Feb 14, 2018
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2 changes: 1 addition & 1 deletion doc/cong.xml
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Expand Up @@ -573,7 +573,7 @@ gap> S := ReesZeroMatrixSemigroup(SymmetricGroup(3),
gap> cong := CongruencesOfSemigroup(S)[3];;
gap> AsSemigroupCongruenceByGeneratingPairs(cong);
<semigroup congruence over <Rees 0-matrix semigroup 2x2 over
Sym( [ 1 .. 3 ] )> with 3 generating pairs>]]></Example>
Sym( [ 1 .. 3 ] )> with 1 generating pairs>]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
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148 changes: 81 additions & 67 deletions gap/congruences/congrms.gi
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Expand Up @@ -1260,106 +1260,120 @@ InstallMethod(GeneratingPairsOfMagmaCongruence,
"for Rees matrix semigroup congruence by linked triple",
[IsRMSCongruenceByLinkedTriple],
function(cong)
local S, g, m, pairs, x, bl, j, rowNo, i, colNo;
local S, G, m, pairs, n_gens, col_pairs, bl, j, row_pairs, max, n, cp, rp,
pair_no, r, c;
S := Range(cong);
g := UnderlyingSemigroup(S);
G := UnderlyingSemigroup(S);
m := Matrix(S);

# Create a list of generating pairs
pairs := [];

# PAIRS FROM THE NORMAL SUBGROUP
# for each x in the subgroup,
# (1,x,1) is related to (1,id,1)
for x in cong!.n do
Add(pairs, [RMSElement(S, 1, x, 1),
RMSElement(S, 1, One(g), 1)]);
od;
# Normal subgroup elements to identify with id
n_gens := GeneratorsOfSemigroup(cong!.n);

# PAIRS FROM THE COLUMNS RELATION
# For each class in the relation...
# Pairs that generate the columns relation
col_pairs := [];
for bl in cong!.colBlocks do
# For each column in the class...
for j in [2 .. Size(bl)] do
# For each row in the matrix...
for rowNo in [1 .. Size(m)] do
Add(pairs,
[RMSElement(S, bl[1], m[rowNo][bl[1]] ^ -1, rowNo),
RMSElement(S, bl[j], m[rowNo][bl[j]] ^ -1, rowNo)]);
od;
Add(col_pairs, [bl[1], bl[j]]);
od;
od;

# PAIRS FROM THE ROWS RELATION
# For each class in the relation...
# Pairs that generate the rows relation
row_pairs := [];
for bl in cong!.rowBlocks do
# For each row in the class...
for i in [2 .. Size(bl)] do
# For each column in the matrix...
for colNo in [1 .. Size(m[1])] do
Add(pairs,
[RMSElement(S, colNo, m[bl[1]][colNo] ^ -1, bl[1]),
RMSElement(S, colNo, m[bl[i]][colNo] ^ -1, bl[i])]);
od;
for j in [2 .. Size(bl)] do
Add(row_pairs, [bl[1], bl[j]]);
od;
od;

# Collate into RMS element pairs
max := Maximum(Length(n_gens), Length(col_pairs), Length(row_pairs));
pairs := EmptyPlist(max);
# Set default values in case a list has length 0
n := One(G);
cp := [1, 1];
rp := [1, 1];
# Iterate through all 3 lists together
for pair_no in [1 .. max] do
# If any list has run out, just keep using the last entry or default value
if pair_no <= Length(n_gens) then
n := n_gens[pair_no];
fi;
if pair_no <= Length(col_pairs) then
cp := col_pairs[pair_no];
fi;
if pair_no <= Length(row_pairs) then
rp := row_pairs[pair_no];
fi;
# Choose r and c s.t. m[r][cp[1]] and m[rp[1]][c] are both non-zero
# (since there are no zeroes, we can just use 1 for both)
r := 1;
c := 1;
# Make the pair and add it
pairs[pair_no] :=
[RMSElement(S, cp[1], m[r][cp[1]] ^ -1 * n * m[rp[1]][c] ^ -1, rp[1]),
RMSElement(S, cp[2], m[r][cp[2]] ^ -1 * m[rp[2]][c] ^ -1, rp[2])];
od;
return pairs;
end);

InstallMethod(GeneratingPairsOfMagmaCongruence,
"for Rees 0-matrix semigroup congruence by linked triple",
[IsRZMSCongruenceByLinkedTriple],
function(cong)
local S, g, m, pairs, i1, x, bl, j, rowNo, i, colNo;
local S, G, m, pairs, n_gens, col_pairs, bl, j, row_pairs, max, n, cp, rp,
pair_no, r, c;
S := Range(cong);
g := UnderlyingSemigroup(S);
G := UnderlyingSemigroup(S);
m := Matrix(S);

# Create a list of generating pairs
pairs := [];

# PAIRS FROM THE NORMAL SUBGROUP
# First, find a matrix entry not equal to zero
i1 := PositionProperty(m[1], x -> x <> 0);
# Normal subgroup elements to identify with id
n_gens := GeneratorsOfSemigroup(cong!.n);

# for each x in the subgroup,
# (i1,x,1) is related to (i1,id,1)
for x in cong!.n do
Add(pairs, [RMSElement(S, i1, x, 1),
RMSElement(S, i1, One(g), 1)]);
od;

# PAIRS FROM THE COLUMNS RELATION
# For each class in the relation...
# Pairs that generate the columns relation
col_pairs := [];
for bl in cong!.colBlocks do
# For each column in the class...
for j in [2 .. Size(bl)] do
# For each row in the matrix...
for rowNo in [1 .. Size(m)] do
if m[rowNo][bl[1]] <> 0 then
Add(pairs,
[RMSElement(S, bl[1], m[rowNo][bl[1]] ^ -1, rowNo),
RMSElement(S, bl[j], m[rowNo][bl[j]] ^ -1, rowNo)]);
fi;
od;
Add(col_pairs, [bl[1], bl[j]]);
od;
od;

# PAIRS FROM THE ROWS RELATION
# For each class in the relation...
# Pairs that generate the rows relation
row_pairs := [];
for bl in cong!.rowBlocks do
# For each row in the class...
for i in [2 .. Size(bl)] do
# For each column in the matrix...
for colNo in [1 .. Size(m[1])] do
if m[bl[1]][colNo] <> 0 then
Add(pairs,
[RMSElement(S, colNo, m[bl[1]][colNo] ^ -1, bl[1]),
RMSElement(S, colNo, m[bl[i]][colNo] ^ -1, bl[i])]);
fi;
od;
for j in [2 .. Size(bl)] do
Add(row_pairs, [bl[1], bl[j]]);
od;
od;

# Collate into RMS element pairs
max := Maximum(Length(n_gens), Length(col_pairs), Length(row_pairs));
pairs := EmptyPlist(max);
# Set default values in case a list has length 0
n := One(G);
cp := [1, 1];
rp := [1, 1];
# Iterate through all 3 lists together
for pair_no in [1 .. max] do
# If any list has run out, just keep using the last entry or default value
if pair_no <= Length(n_gens) then
n := n_gens[pair_no];
fi;
if pair_no <= Length(col_pairs) then
cp := col_pairs[pair_no];
fi;
if pair_no <= Length(row_pairs) then
rp := row_pairs[pair_no];
fi;
# Choose r and c s.t. m[r][cp[1]] and m[rp[1]][c] are both non-zero
r := First([1 .. Length(m)], r -> m[r][cp[1]] <> 0);
c := First([1 .. Length(m[1])], c -> m[rp[1]][c] <> 0);
# Make the pair and add it
pairs[pair_no] :=
[RMSElement(S, cp[1], m[r][cp[1]] ^ -1 * n * m[rp[1]][c] ^ -1, rp[1]),
RMSElement(S, cp[2], m[r][cp[2]] ^ -1 * m[rp[2]][c] ^ -1, rp[2])];
od;
return pairs;
end);

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8 changes: 4 additions & 4 deletions tst/standard/conguniv.tst
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Original file line number Diff line number Diff line change
Expand Up @@ -238,10 +238,10 @@ gap> S := Semigroup([Transformation([2, 1, 2]),
gap> uni := UniversalSemigroupCongruence(S);
<universal semigroup congruence over <transformation semigroup of degree 3
with 2 generators>>
gap> GeneratingPairsOfSemigroupCongruence(uni);
[ [ Transformation( [ 1, 2, 1 ] ), Transformation( [ 1, 2, 1 ] ) ],
[ Transformation( [ 2, 1, 2 ] ), Transformation( [ 1, 2, 1 ] ) ],
[ Transformation( [ 1, 2, 1 ] ), Transformation( [ 1, 2, 2 ] ) ] ]
gap> pairs := GeneratingPairsOfSemigroupCongruence(uni);;
gap> cong := SemigroupCongruenceByGeneratingPairs(S, pairs);;
gap> NrCongruenceClasses(cong);
1

#T# IsUniversalSemigroupCongruence for a cong by generating pairs
gap> S := Semigroup([PartialPerm([1], [2]),
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